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ION–ION INTERACTIONS 315

values of dielectric constant are small. In such solutions of electrolytes therefore it has
already been stated that ion-pair formation is favored.
Suppose that the electrostatic forces are sufficiently strong then it may well
happen that the ion-pair “dipoles” may attract ions and triple ions be formed; thus

or

Charged triple ions have been formed from uncharged ion pairs. These charged triple
ions play a role in determining activity coefficients. Triple-ion formation has been
suggested in solvents for which < 15. The question of triple-ion formation can be
treated on the same lines as those for ion-pair formation.
A further decrease of dielectric constant below a value of about 10 may make
possible the formation of still larger clusters of four, five, or more ions. In fact, there
is some evidence for the clustering of ions into groups containing four ions in solvents
of low dielectric constant.

3.9. THE VIRIAL COEFFICIENT APPROACH TO DEALING WITH
SOLUTIONS

The material so far presented has shown that a model taking into account the size
of the ions, ion-pair formation, and the idea that some of the water in the solution was
not to be counted in estimating the “effective” concentration—the activity—has
allowed a fair accounting for the main arbiter of the interionic attraction energy, the
activity coefficient, even at concentrations up to nearly 5 mol (see, e.g., Fig.
3.39).
24
This position of ionic solution theory has, however, a challenger, and, during
the 1970s and 1980s, it was this radically different approach to ionic solution theory

24
There might be some who would actually say that there was “something wrong” with the theory of Debye
and Hückel, but this claim depends on which version of the theory one means. The limiting-law equation
certainly is inconsistent with experiments above for a 1:1 electrolyte and even smaller
concentrations for electrolytes possessing ions with a valence above unity. The later developments of the
theory, which take into account the space occupied by the ions, do much better and taking the effect of
solvation into account gives agreement with experiment to concentrations up to (Fig. 3.39).
One question relates to whether cations and anions should have the same activity coefficient (the simple,
original Debye and Hückel theory predicts this), but if one extends the model to account for “dead water”
around ions, it turns out that there is more of this with small cations (they cling to water tighter) than with
big anions, where the ion–water electric field is less and hence adherent dead waters are less in number.
This solvational difference would imply a higher activity coefficient at high concentrations for cations
than for anions, for which there is some evidence.